non-equilibrium dynamics across impurity quantum critical ... · non-equilibrium dynamics across...
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Non-Equilibrium Dynamics across Impurity Quantum Critical
Points
Marco Schiro’Princeton Center For Theoretical Science
M. Schiro’, Phys. Rev. B 86 161101(R) (2012)
KITP Program on Dynamics in Closed Thermally Isolated Systems, S. Barbara October 2012
Thursday, June 27, 13
Outline
Thursday, June 27, 13
Outline
Motivations (Theory): Local Quenches in Impurity Models
Thursday, June 27, 13
Outline
Motivations (Experiment): Kondo Effect in the Optical Spectrum of Quantum Dots
Motivations (Theory): Local Quenches in Impurity Models
Thursday, June 27, 13
Outline
Basic Questions, Quick Answers and Few Examples
Motivations (Experiment): Kondo Effect in the Optical Spectrum of Quantum Dots
Motivations (Theory): Local Quenches in Impurity Models
Thursday, June 27, 13
Outline
Basic Questions, Quick Answers and Few Examples
Motivations (Experiment): Kondo Effect in the Optical Spectrum of Quantum Dots
A Less Trivial (?) Example -- The Role of Kondo Effect
Motivations (Theory): Local Quenches in Impurity Models
Thursday, June 27, 13
Small quantum systems coupled to a “bath’’
Few interacting quantum degrees of freedom
Continuum of Excitations
Example: Anderson Impurity Model
Vk
!(")
"!W W
Bath
Quantum “Impurity’’
H =U
2(n� 1)2 +
X
k�
�k f†k� fk� +
X
k�
Vk
�c†� fk� + h.c.
�
Thursday, June 27, 13
Dynamics after Local Quenches
Long time dynamics after a local perturbation: does the system thermalize?
hOit = Tr[�0 eiHt O e�iHt] t ! 1 hOit = hOieq ???
Vk
t = 0 t > 0Bath Bath
Non trivial (universal) off equilibrium dynamics?
Also relevant for closed systems through DMFT mapping
See recent works: Ratiani & Mitra, PRB 81 125110(2010), Vinkler, Schiller & Andrei PRB 85 035411 (2012), ...
Thursday, June 27, 13
Experimental Motivations
Diluted Magnetic Impurities in Metals ~ 1960
Quantum Dots and Single Molecule Devices ~1998
Optically controlled Self-Assembled Quantum Dots
~ 2005
Vg
self-organised quantum dots
InAs/GaAs
laser
2DEG
Thursday, June 27, 13
Experimental Motivations
Diluted Magnetic Impurities in Metals ~ 1960
Quantum Dots and Single Molecule Devices ~1998
Optically controlled Self-Assembled Quantum Dots
~ 2005
Vg
self-organised quantum dots
InAs/GaAs
laser
2DEG
Thursday, June 27, 13
New All Optical Set-up: Self Assembled QuantumDots
Vg
self-organised quantum dots
back contact
top gate
InAs/GaAslaserz
0
ωL
2DEG
Absorption/Emission spectrum of a single Quantum Dot!
Evidence of Strong Hybridization to the Fermi Sea
Karrai et al, Nature (2004), Dalgarno et al, PRL (2008)
Warburton et al., Nature 405, 926 (2000)
Thursday, June 27, 13
Light-Induced Local “Quantum Quench’’
initial final
Helmes et al, PRB 72 125301 (2005)Tureci et al, PRL 106 107402 (2011)Munder et al, PRB 85 235104 (2012)
A�(⌅L) = 2⇥X
nm
⇤im |⇥n; f | e†�|m; i⇤|2 �(⌅L � Efn + Ei
m)
Work Statistics!Silva, PRL 101 120603 (2008)
Heyl&Kehrein, PRL 108 190601 (2012)
Pump-Probe Dynamics (soon??)
HL ⇠ e†� h†� e
�i⇥L t + h.c.
Hdot =X
�
(�e� ne� + �h� nh�)+
+U ne" ne# �X
��0
Uehne� nh�0
Optical Absorption Spectrum
Thursday, June 27, 13
Connection with Work Statistics Heyl&Kehrein, PRL 108 190601 (2012)
Sudden Local Quench: Hi ! Hf �H ⇠ Ueh
X
�
ne�
Non-equilibrium Correlators
PB(W ) =
Zdt
2⇥eiWt h eiHf te�iHitif ⇠ E(� = W )
PF (W ) =
Zdt
2⇥eiWt h eiHi te�iHf tii ⇠ A(� = W )
Crooks Relation
Absorption!
Emission!
A(�)
E(�)⇠ e� (⇥��F )
Hi Hf
NB: Also in a single shot, for weak perturbationThursday, June 27, 13
Kondo Effect in the Optical Absorption Spectrum
C. Latta et al., Nature 474, 627 (2011)
Edge Singularity in the Optical Absorption Spectrum!
A(�) ⇠ ��⌘
Exponent tunable by magnetic field/gate voltage
Thursday, June 27, 13
Toward Hybrid Light-Matter Quantum Impurity Systems
Coupling Electrons with quantum light (Photons)Response of Kondo State to microwave signals
Delbecq et al , PRL (2011) ENS-LPA (Kontos Lab)
Thursday, June 27, 13
Theoretical Question & Quick Answers
Long time dynamics after a local perturbation: does the system thermalize? How?
hOit = Tr[�0 eiHt O e�iHt] t ! 1 hOit = hOieq ???
Vk
t = 0 t > 0Bath Bath
Thursday, June 27, 13
Theoretical Question & Quick Answers
Long time dynamics after a local perturbation: does the system thermalize? How?
hOit = Tr[�0 eiHt O e�iHt] t ! 1 hOit = hOieq ???
Vk
t = 0 t > 0Bath Bath
YES!
• Conserved quantities (energy, particle number, spin,..) flow across the system due to • Reservoir is infinite and gapless (and if proper order of limit is taken )
Vk
Thursday, June 27, 13
Theoretical Question & Quick Answers
Long time dynamics after a local perturbation: does the system thermalize? How?
hOit = Tr[�0 eiHt O e�iHt] t ! 1 hOit = hOieq ???
Vk
t = 0 t > 0Bath Bath
YES! NO!
• Conserved quantities (energy, particle number, spin,..) flow across the system due to • Reservoir is infinite and gapless (and if proper order of limit is taken )
Vk
• Anderson Impurity Model (or similar)
is integrable -- Bethe Ansatz Solvable
Thursday, June 27, 13
Example (simple): Electrons coupled to a Fermi Sea
H = �0X
�
c†� c� +X
k�
�k f†k� fk� +
X
k�
Vk
�c†� fk� + h.c.
�
Model is quadratic, solution easy!
Long time dynamics: relaxation to the ground state
Bath
✏0
W
�W
0
� = ⇤X
k
V 2k �(⇥k)
t � 1/�Thursday, June 27, 13
End of the story??
Thursday, June 27, 13
End of the story??
Not Yet ..!!
Thursday, June 27, 13
End of the story??
•We are assuming system+bath always ‘effectively’ coupled
•Local Many Body Interactions can change this picture?
Not Yet ..!!
Thursday, June 27, 13
Anderson impurity model...
Anderson, Kondo, Haldane, Nozieres, Wilson,...(~1960-1975)
T
FO
LM
SC
U
TK
Free Orbital FP:Charge Fluctuations
Local Moment FP:Spin Fluctuations
Strong Couplig FP:Kondo “Screening”
H =U
2(n� 1)2 +
X
k�
�k f†k� fk� +
X
k�
Vk
�c†� fk� + h.c.
�
�eff
SC
LMFO
LM 0
|0⇤, | �⇤, | ⇥⇤, | �⇥⇤
| �⇤, | ⇥⇤
(| �⇤| ⇥⇤+ | ⇥⇤| �⇤)/⌅2
Thursday, June 27, 13
... and its out of equilibrium dynamics
TD-NRG: Anders&Schiller PRL(2005)
• Charge relaxation time
• Spin relaxation time tspin ⇠ 1/TK
tcharge ⇠ 1/�
Thursday, June 27, 13
... and its out of equilibrium dynamics
Take-home message:Kondo Effect, always thermal, but RG flow matters!
TD-NRG: Anders&Schiller PRL(2005)
• Charge relaxation time
• Spin relaxation time tspin ⇠ 1/TK
tcharge ⇠ 1/�
Thursday, June 27, 13
Ferromagnetic Kondo Model
Quench Dynamics from a decoupled initial state
J? ! 0
J?(�) = J/(1 + �F J log(�/D))
Hackl et al, PRL 102 196101 (2009)Flow-Equation, tdNRG
Relaxation, yet very slowMemory of Initial Condition
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Relevant Examples:
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Relevant Examples:
Two Impurity Kondo Model
Pseudo-Gap Anderson Model
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Relevant Examples:
Two Impurity Kondo Model
Pseudo-Gap Anderson Model
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
Fradkin, Ingersent, Vojta, ...
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Relevant Examples:
Two Impurity Kondo Model
Pseudo-Gap Anderson Model
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
Fradkin, Ingersent, Vojta, ...
Jones&Varma, Jones,Millis&Kotliar, Affleck,Ludwing&Jones,..
�eff (t) ! 0
Thursday, June 27, 13
To Screen or Not to Screen...
Kondo Screening
No Kondo Screening
ggc
Relevant Examples:
Two Impurity Kondo Model
Pseudo-Gap Anderson Model
Flow to Strong Coupling FPThermalization at long times
screenedsinglet
Flow to Local Moment FP
What happens at long times??
How these dynamical regimes are connected?
Fradkin, Ingersent, Vojta, ...
Jones&Varma, Jones,Millis&Kotliar, Affleck,Ludwing&Jones,..
�eff (t) ! 0
Thursday, June 27, 13
Anderson Impurity in a power-law bath
H =U
2(n� 1)2 +
X
k�
�k f†k� fk� +
X
k�
Vk
�c†� fk� + h.c.
�
AIM in a pseudo-gapFradkin, Ingersent, Vojta,Polkovnikov,..
✏
�(✏) � |✏|r
�(✏)
U/�SC LM
Quantum Critical Point!r < 1/2
No QCP for r > 1/2
UcThursday, June 27, 13
Out of Equilibrium Dynamics
Initial Condition
H =U
2(n� 1)2 +
X
k�
�k f†k� fk� +
X
k�
Vk
�c†� fk� + h.c.
�
No Coupling with Bath
No Local Interaction
Real-Time Dynamics: how to solve?
Vk = 0
U = 0 �(✏) � |✏|r
�(✏)
time-dep NRG
diagrammatic MC
time-dep variational method
Finite Size Effects due to Wilson Chain
cfr A. Rosch, Eur. Phys. J. B 85 6 (2012)
dE
dT
⇠ 1
log�
dN
dµ⇠ N
Thursday, June 27, 13
Dynamics with Real-Time QMCMS, PRB 81, 103232 (2010)
• Seemingly slow dynamics for large but...
• Short time dynamics only, due to “sign’’ problem
• Finite Temperature in the bath, no QCP!
r > 1/2
Thursday, June 27, 13
A Time Dependent Variational Approach
|⇥(t)⇥ = eiHt |⇥(0)⇥ � P (t) |�(t)⇥
Ansatz on the time dep Many Body Wave Function
Variational parameters:
�
Zdt ⇥�(t)| i⇥t �H|�(t)⇤ = 0
P (t) =X
a=|0⇤,..,|�⇥⇤
�a(t) ei�a(t) | a⇥� a|
Gutzwiller WF:change weight of atomic
configurations
Non interacting model with Vk(t)i�t|�(t)� = H�(t)|�(t)�
H⇥(t) =X
k�
�k f†k� fk� +
pZ(t)
X
k�
Vk
�c†� fk� + h.c.
�
Thursday, June 27, 13
A Time Dependent Variational Approach
D = 4 Ehyb(t)p
D (1/2�D) sin�
� = U + Ehyb(t)1� 4Dp
D (1/2�D)cos�
Dynamics of Variational Parameters
D(t),�(t)
Ehyb(t) =X
k�
Vk ��(t)|⇣f†k� c� + h.c.
⌘|�(t)⇥
i�t|�(t)� = H�(t)|�(t)�
Dynamics of non-equilibrium bath
H⇥(t) =X
k�
�k f†k� fk� +
pZ(t)
X
k�
Vk
�c†� fk� + h.c.
�
Thursday, June 27, 13
Variational Ground State
A QCP between Kondo and Local moment regime
Uc/� =16(1 + r)
� r
NB: Uc ! 1 for r ! 0
|⇥⇥ =
0
@X
a=|0⇤,..,|�⇥⇤
�a | a⇥� a|
1
A |�⇥
H⇥ =X
k�
�k f†k� fk� +
�ZX
k�
Vk
�c†� fk� + h.c.
�
Ground-State of H?
E[�a,⇤Z] = ��|H|�⇥
Variational Energy
0 10 20 30 40 50U/K
0.001
0.01
0.1
1
Z
r = 0.0r = 0.1r = 0.2r = 0.3r = 0.4
0 0.2 0.4 0.6 0.8 1r
50
100
150
200
Uc/K
Uc(r)
Z~exp(-/U/16K)
Thursday, June 27, 13
Dynamics in the Kondo Screened Phase
0 10
-0.5
0
0.5
!(t)
0 5 10 150
0.2
0.4
0.6
0.8
1
Z(t)
0 5 10 15 20
t"
-3
-2.5
-2
-1.5
-1
-0.5
0
Eh(t)
0 5 10 15
t"
0
0.05
0.1
0.15
0.2
0.25
D(t)
0 5 10 150
0.05
0.1
0.15
0.2
0.25
D(t)
Relaxation to the thermal steady state
U < Uc(r)r = 0.4
Thursday, June 27, 13
Dynamics in the Local Moment Phase
No Damping, No Relaxation, No Steady State
0 10 20 30 40 50 600
200400600800
(t)
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
Z(t)
0 5 10 15 20
tK-3
-2
-1
0
Eh(t)
0 5 10 15 20
U/K0
0.5
1
1.5
2-_/10d
0 10 20 30 40 50 600
0.050.10.150.20.250.3
D(t)
0 2 4 6 8 10-0.03
0
0.03
Eh(t)
U > Uc(r)
Thursday, June 27, 13
How these regimes are connected?
In the Kondo Phase, linearize dynamics around steady state
Relaxation time diverges at the quantum critical point!
� ⇠ E�hyb E�
hyb ⇠p
Z�
Equilibrium Properties:
⇥ O = �⇥O + ⇤⇥Ehyb�O = O(t)�O?
� ⇠ �E�2hyb/Z�
Dynamical transition between thermal and non thermal regime
�rel ⇥ 1/pZ� ⇥ |U � Uc(r)|�1/2
Thursday, June 27, 13
Conclusions• Thermalization in Quantum Impurity Models after local
perturbations
• Long-Time physics is sensitive to Low-Energy features, i.e. RG Fixed Points, Kondo Effect
• Non trivial Dynamics across a Kondo-to-Local Moment QCP
Open Questions:
Role of Quantum Fluctuations: damping in the LM?Correlation Functions (Aging?), Statistics of Work
Other Models: 2IKM (Bosonization?),...Thursday, June 27, 13
Thanks!
Thursday, June 27, 13
Experimental Setup
Helmes et al, PRB 72 125301 (2005)Latta et al, Nature 474 627 (2011)
InAs/GaAs Tunnel Barriers result into dips in conduction/valence band gap: quantum dots!
Number of electrons tunable by Vg
Light excites e-h pairs that recombine emitting photons
Thursday, June 27, 13
Excitonic Anderson ModelH = HQD +Hbath +Hhyb
HQD =X
�
�c nc� + Uc nc" nc#+
+X
�
�v nv� + Uv (1� nv") (1� nv#)
�X
��0
Uexc nc�(1� nv�0)Exciton Binding Energy: e-h attraction
Helmes et al, PRB 72 125301 (2005)
HLM =X
k�
(�k ake�i⇥kt c†�v� + hc)
Light-Matter Interaction: absorption vs emission
Thursday, June 27, 13